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normed vector space

Table of Contents

1. Introduction

A normed vector space is a vector space with a norm defined, which describes the "length" of the vector. This norm obeys these properties:

\begin{align} \label{} \lVert ax \rVert = \lvert a \rvert \lVert x \rVert \\ \lVert x + y \rVert \le \lVert x \rVert + \lVert y \rVert \end{align}

this gives rise to a metric \(d(x, y)\):

\begin{align} \label{} d(x, y) = \lVert x - y \rVert \end{align}
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